Metastability for the Ising model on a random graph
Metastability for the Ising model
on a random graph
Joint project with
Sander Dommers, Frank den Hollander, Francesca Nardi
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Outline
• A little about metastability
• Some key questions
• A little about the Ising model on a multi-graph, and the
Configuration model random graph
• Significance of low temperature limit
• The communication height and some of its properties
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Concept of metastability
We observe (Markovian) dynamics on some state space Ω with
energy function H that has a global minimum s and or more local
minima m.
Dynamics started at m may appear to be in equilibrium, prevented
from moving far due to a
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Concept of metastability
We observe (Markovian) dynamics on some state space Ω with
energy function H that has a global minimum s and or more local
minima m.
Dynamics started at m may appear to be in equilibrium, prevented
from moving far due to a
potential barrier.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
m and s
• s is a stable state (trivially unique in our model).
Metastability for the Ising model on a random graph
Ising model on a multi-graph
m and s
• s is a stable state (trivially unique in our model).
• m is a metastable state if no other state has a greater potential
barrier between itself and configurations of lower energy (along
best path).
V (m) ≥ V (q)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Main questions
What can we say about the crossover time
m→s?
In particular,
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Main questions
What can we say about the crossover time
m→s?
In particular,
• what is the average crossover time Em [τs ]?
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Main questions
What can we say about the crossover time
m→s?
In particular,
• what is the average crossover time Em [τs ]?
• how does it vary with |Ω| ?
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Main questions
What can we say about the crossover time
m→s?
In particular,
• what is the average crossover time Em [τs ]?
• how does it vary with |Ω| ?
• what kind of configurations does the dynamics see ”near the
top”?
We investigate these in the low temperature setting β → ∞ (this
is a parameter of the model).
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Ising model on a multi-graph
Let G = (V, E) be a multi-graph.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Ising model on a multi-graph
Let G = (V, E) be a multi-graph.
• Configuration (state) space : Ω = {+1, −1}|V | is the set of
+1/ − 1 configurations on G.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Ising model on a multi-graph
Let G = (V, E) be a multi-graph.
• Configuration (state) space : Ω = {+1, −1}|V | is the set of
+1/ − 1 configurations on G.
• Hamiltonian on configuration space: for ω ∈ Ω,
H (ω) = −
J
2
with J > 0 and h > 0.
X
<u,v >∈E
ω (v ) ω (u) −
hX
ω (v )
2
v ∈V
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Ising model on a multi-graph
Let G = (V, E) be a multi-graph.
• Configuration (state) space : Ω = {+1, −1}|V | is the set of
+1/ − 1 configurations on G.
• Hamiltonian on configuration space: for ω ∈ Ω,
H (ω) = −
J
2
X
ω (v ) ω (u) −
<u,v >∈E
hX
ω (v )
2
v ∈V
with J > 0 and h > 0.
• Gibbs measure on Ω
µβ (ω) =
1
exp {−βH (ω)}
Zβ
Assigns exponential preference to configurations with low energy.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Configuration model random graph
A family of random graphs that are of particular interest to us, are
based on the Configuration model. This begins with a given set of
vertices {v1 , ..., vn }, and a prescribed degree sequence
d = (d1 , ..., dn ).
E.g. for the example below, d = (4, 4, 3, 4, 4, 3)
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Glauber Dynamics, significance of β → ∞
Dynamics on configuration space: continuous-time Glauber
dynamics with transition rates
(
if ω v ω0
exp −β [H (ω0) − H (ω)]+
cβ (ω, ω0) =
0
otherwise
where ω v ω0 iff ω and ω0 differ at exactly one vertex.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Glauber Dynamics, significance of β → ∞
Dynamics on configuration space: continuous-time Glauber
dynamics with transition rates
(
if ω v ω0
exp −β [H (ω0) − H (ω)]+
cβ (ω, ω0) =
0
otherwise
where ω v ω0 iff ω and ω0 differ at exactly one vertex.
• Note that ’up’ moves become exponentially expensive.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Glauber Dynamics, significance of β → ∞
Dynamics on configuration space: continuous-time Glauber
dynamics with transition rates
(
if ω v ω0
exp −β [H (ω0) − H (ω)]+
cβ (ω, ω0) =
0
otherwise
where ω v ω0 iff ω and ω0 differ at exactly one vertex.
• Note that ’up’ moves become exponentially expensive.
• Bounds on Reff (effective resistance) show that w.h.p. dynamics
avoid ” unnecessarily expensive” configurations when going from
m → s , and will take an optimal path (see for example
Metastability - A. Bovier, F. den Hollander).
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
The communication height between m and s on G is
Γ? := Φ (m, s) − H (m)
=
min max H (σ) − H (m)
γ:m→s σ∈γ
- represents the ”lowest barrier” between m and s which the
dynamics must overcome.
- paths γ that correspond to the minimum are called optimal paths.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Communication Height
For example, for the previous graph we have the following optimal
path for suitable values of J and h (note: easy to show here that
m = and s = )
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Significance of Γ?
How is Γ? related to the crossover time m → s? It follows from
standard universal metastability theorems that
lim exp (−βΓ? ) E [τ ] = K
β→∞
Furthermore, in T2 ⊂ Z2 the pre-factor K is related to |C ? |, where
C ? := Configurations on optimal paths where height Γ? is first seen.
We expect that a similar relation holds for our model.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
When mini di ≥ 3, the resulting Configuration model is an
expander graph.
Therefore Γ? = Θ (n), with n = |V | (under some assumptions on
the distribution of d).
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
When mini di ≥ 3, the resulting Configuration model is an
expander graph.
Therefore Γ? = Θ (n), with n = |V | (under some assumptions on
the distribution of d).
Upper bound on Γ? : by a coupling argument, we can show that
Γ? ≤ `n /4
where `n is the total degree of the graph. We would like to make
this sharper.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
We also expect that
Γ?n
→C
n
for some constant C depending on the parameters of the model,
when d is regular, and possibly also for di ∼ i.i.d. under some
constraints.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
We also expect that
Γ?n
→C
n
for some constant C depending on the parameters of the model,
when d is regular, and possibly also for di ∼ i.i.d. under some
constraints.
To this avail, we can show that if A ⊂ V is a random
set/configuration of total degree `A , then
`A − 1
|∂A|
→L2 `A 1 −
|d|
|d| − 1
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
`A − 1
|∂A|
→L2 `A 1 −
|d|
|d| − 1
We need to show that this holds not only for random sets, but also
for extremal.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Some properties of Γ?
`A − 1
|∂A|
→L2 `A 1 −
|d|
|d| − 1
We need to show that this holds not
only for random sets, but also for extremal.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
back to m and s
But what are m and s?
It is easy to see that s = , i.e. the configuration with +1
assigned at every vertex.
It is also easy to see that is a local minimum (for
non-degenerate values of J and h).
Showing that m = and nothing else (in the case when degree
sequence satisfies di ≥ 3) is difficult.
Metastability for the Ising model on a random graph
Ising model on a multi-graph
Conclusion
• There are many unforeseen challenges when assigning the Ising
model to a random graph.
• Global properties of the random graph model play a key role.
• Precise descriptions of Γ? , C ? exist in lattice setting, but may be
too difficult for random graphs. But good estimates should be
attainable.